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 Cos

 http://functions.wolfram.com/01.07.21.2721.01

 Input Form

 Integrate[z^n E^(p z^2) Sin[b z] Cos[c z]^v, z] == (-I) 2^(-2 - v) p^(-1 - n) (E^(b^2/(4 p)) Binomial[v, v/2] (-1 + Mod[v, 2]) Sum[2^(-n + q) ((-I) b)^(n - q) ((b - 2 I p z)^2/p)^((1/2) (-1 - q)) (I b + 2 p z)^(1 + q) Binomial[n, q] Gamma[(1 + q)/2, (b - 2 I p z)^2/(4 p)], {q, 0, n}] - E^(b^2/(4 p)) Binomial[v, v/2] (-1 + Mod[v, 2]) Sum[2^(-n + q) (I b)^(n - q) ((b + 2 I p z)^2/p)^ ((1/2) (-1 - q)) ((-I) b + 2 p z)^(1 + q) Binomial[n, q] Gamma[(1 + q)/2, (b + 2 I p z)^2/(4 p)], {q, 0, n}] + p^(1/2 + n) Sum[E^((b^2 + c^2 (-2 s + v)^2)/(2 p)) Binomial[v, s] ((-E^(-((b - 2 c s + c v)^2/(4 p)))) Sum[2^(-n + q) p^(-(1/2) - n) ((-I) (b + 2 c s - c v))^(n - q) (I (b + 2 c s - c v - 2 I p z))^ (1 + q) ((b + 2 c s - c v - 2 I p z)^2/p)^((1/2) (-1 - q)) Binomial[n, q] Gamma[(1 + q)/2, (b + 2 c s - c v - 2 I p z)^2/ (4 p)], {q, 0, n}] - Sum[2^(-n + q) p^(-(1/2) - n) ((-I) (b - 2 c s + c v))^(n - q) (I (b - 2 c s + c v - 2 I p z))^ (1 + q) ((b - 2 c s + c v - 2 I p z)^2/p)^((1/2) (-1 - q)) Binomial[n, q] Gamma[(1 + q)/2, (b - 2 c s + c v - 2 I p z)^2/ (4 p)], {q, 0, n}]/E^((b + 2 c s - c v)^2/(4 p)) + Sum[2^(-n + q) p^(-(1/2) - n) (I (b + 2 c s - c v))^(n - q) ((-I) (b + 2 c s - c v + 2 I p z))^(1 + q) ((b + 2 c s - c v + 2 I p z)^2/p)^((1/2) (-1 - q)) Binomial[n, q] Gamma[(1 + q)/2, (b + 2 c s - c v + 2 I p z)^2/(4 p)], {q, 0, n}]/ E^((b - 2 c s + c v)^2/(4 p)) + Sum[2^(-n + q) p^(-(1/2) - n) (I (b - 2 c s + c v))^(n - q) ((-I) (b - 2 c s + c v + 2 I p z))^(1 + q) ((b - 2 c s + c v + 2 I p z)^2/p)^((1/2) (-1 - q)) Binomial[n, q] Gamma[(1 + q)/2, (b - 2 c s + c v + 2 I p z)^2/(4 p)], {q, 0, n}]/ E^((b + 2 c s - c v)^2/(4 p))), {s, 0, Floor[(1/2) (-1 + v)]}]) /; Element[v, Integers] && v > 0 && Element[n, Integers] && n >= 0

 Standard Form

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 MathML Form

 z n p z 2 sin ( b z ) cos v ( c z ) z - 2 - v - 2 p - n - 1 ( ( s = 0 v - 1 2 b 2 + c 2 ( v - 2 s ) 2 2 p ( v s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox["s", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( - ( b + 2 c s - c v ) 2 4 p q = 0 n 2 q - n p - n - 1 2 ( ( b - 2 c s + c v ) ) n - q ( - ( b - 2 c s + c v + 2 p z ) ) q + 1 ( ( b - 2 c s + c v + 2 p z ) 2 p ) 1 2 ( - q - 1 ) ( n q ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["q", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] Γ ( q + 1 2 , ( b - 2 c s + c v + 2 p z ) 2 4 p ) + - ( b - 2 c s + c v ) 2 4 p q = 0 n 2 q - n p - n - 1 2 ( ( b + 2 c s - c v ) ) n - q ( - ( b + 2 c s - c v + 2 p z ) ) q + 1 ( ( b + 2 c s - c v + 2 p z ) 2 p ) 1 2 ( - q - 1 ) ( n q ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["q", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] Γ ( q + 1 2 , ( b + 2 c s - c v + 2 p z ) 2 4 p ) - - ( b + 2 c s - c v ) 2 4 p q = 0 n 2 q - n p - n - 1 2 ( - ( b - 2 c s + c v ) ) n - q ( ( b - 2 c s + c v - 2 p z ) ) q + 1 ( ( b - 2 c s + c v - 2 p z ) 2 p ) 1 2 ( - q - 1 ) ( n q ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["q", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] Γ ( q + 1 2 , ( b - 2 c s + c v - 2 p z ) 2 4 p ) - - ( b - 2 c s + c v ) 2 4 p q = 0 n 2 q - n p - n - 1 2 ( - ( b + 2 c s - c v ) ) n - q ( ( b + 2 c s - c v - 2 p z ) ) q + 1 ( ( b + 2 c s - c v - 2 p z ) 2 p ) 1 2 ( - q - 1 ) ( n q ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["q", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] Γ ( q + 1 2 , ( b + 2 c s - c v - 2 p z ) 2 4 p ) ) ) p n + 1 2 - b 2 4 p ( v v 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox[FractionBox["v", "2"], Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( v mod 2 \$CellContext`v 2 - 1 ) q = 0 n 2 q - n ( b ) n - q ( ( b + 2 p z ) 2 p ) 1 2 ( - q - 1 ) ( - b + 2 p z ) q + 1 ( n q ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["q", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] Γ ( q + 1 2 , ( b + 2 p z ) 2 4 p ) + b 2 4 p ( v v 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox[FractionBox["v", "2"], Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( v mod 2 \$CellContext`v 2 - 1 ) q = 0 n 2 q - n ( - b ) n - q ( ( b - 2 p z ) 2 p ) 1 2 ( - q - 1 ) ( b + 2 p z ) q + 1 ( n q ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["q", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] Γ ( q + 1 2 , ( b - 2 p z ) 2 4 p ) ) /; v + n Condition z z n p z 2 b z c z v -1 2 -1 v -2 p -1 n -1 s 0 v -1 2 -1 b 2 c 2 v -1 2 s 2 2 p -1 Binomial v s -1 b 2 c s -1 c v 2 4 p -1 q 0 n 2 q -1 n p -1 n -1 1 2 b -1 2 c s c v n -1 q -1 b -1 2 c s c v 2 p z q 1 b -1 2 c s c v 2 p z 2 p -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 b -1 2 c s c v 2 p z 2 4 p -1 -1 b -1 2 c s c v 2 4 p -1 q 0 n 2 q -1 n p -1 n -1 1 2 b 2 c s -1 c v n -1 q -1 b 2 c s -1 c v 2 p z q 1 b 2 c s -1 c v 2 p z 2 p -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 b 2 c s -1 c v 2 p z 2 4 p -1 -1 -1 b 2 c s -1 c v 2 4 p -1 q 0 n 2 q -1 n p -1 n -1 1 2 -1 b -1 2 c s c v n -1 q b -1 2 c s c v -1 2 p z q 1 b -1 2 c s c v -1 2 p z 2 p -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 b -1 2 c s c v -1 2 p z 2 4 p -1 -1 -1 b -1 2 c s c v 2 4 p -1 q 0 n 2 q -1 n p -1 n -1 1 2 -1 b 2 c s -1 c v n -1 q b 2 c s -1 c v -1 2 p z q 1 b 2 c s -1 c v -1 2 p z 2 p -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 b 2 c s -1 c v -1 2 p z 2 4 p -1 p n 1 2 -1 b 2 4 p -1 Binomial v v 2 -1 \$CellContext`v 2 -1 q 0 n 2 q -1 n b n -1 q b 2 p z 2 p -1 1 2 -1 q -1 -1 b 2 p z q 1 Binomial n q Gamma q 1 2 -1 b 2 p z 2 4 p -1 b 2 4 p -1 Binomial v v 2 -1 \$CellContext`v 2 -1 q 0 n 2 q -1 n -1 b n -1 q b -1 2 p z 2 p -1 1 2 -1 q -1 b 2 p z q 1 Binomial n q Gamma q 1 2 -1 b -1 2 p z 2 4 p -1 v SuperPlus n [/itex]

 Rule Form

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 Date Added to functions.wolfram.com (modification date)

 2002-12-18